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Thursday, July 31, 2014

Homogeneous Functions - Arbitrary Constant, 3

Category: Differential Equations

"Published in Vacaville, California, USA"

Find the particular solution for


in which y = 3 when x = 1.

Solution:

Consider the given equation above


Did you notice that the given equation cannot be solved by separation of variables? The algebraic functions are the combination of x and y in the group and there's no way that we can separate x and y.  

This type of differential equation is a homogeneous function. Let's consider this procedure in solving the given equation as follows   
 
 



Let


so that


Substitute the values of y and dy to the given equation, we have   






The resulting equation can now be separated by separation of variables as follows   




Integrate on both sides of the equation, we have  







Take the inverse natural logarithm on both sides of the equation, we have




But



Hence, the above equation becomes 





Substitute the value of x and y in order to get the value of C as follows 






Therefore, the particular solution is



 

There's another way in getting the particular solution of the given equation above as follows





Integrate both sides of the equation, we have  






Substitute the value of x and y in order to get the value of C as follows 





Therefore, the particular solution is